A201637 → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(-18\,x^{2} + 18\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(3\,x^{2} - 3\,x^{3} - 2\,n x^{2} + 2\,n x^{3}\right)\,P_{n-2}'(x) + \left(1 - 8\,x + 4\,n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)

\(P_{1}(x) = 1\)

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1	1														1
2	1														1
3	1	4													5
4	1	16	18												35
5	1	44	174	96											315
6	1	104	984	1776	600										3465
7	1	228	4356	17600	18540	4320									45045
8	1	480	16830	128480	290700	203904	35280								675675
9	1	988	59850	783200	3220100	4711824	2387952	322560							11486475
10	1	2008	201780	4250000	28827500	74640384	77184072	29871360	3265920						218243025
11	1	4052	656544	21304640	223825000	932144304	1672334328	1297495296	399051360	36288000					4583103525
12	1	8144	2085114	100975616	1571848600	9890305824	28018865616	37137311616	22567330944	5683564800	439084800				105411381075
13	1	16332	6511878	459505504	10257413880	93293764704	392289791488	809443464192	829798112016	407883018240	86104909440	5748019200			2635284526875
14	1	32712	20099088	2029037104	63349088880	805940521344	4814040324688	14545509138432	22948789260816	18830796618240	7677223804800	1384033029120	80951270400		71152682225625
15	1	65476	61523964	8759319232	375080261140	6508059074496	53443825045744	226399280528896	517449361181328	647283751398720	436585358613888	150610848215040	23542903238400	1220496076800	2063427784543125

Technical details

Node id: n-8e982d8b673fb1df · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-fbfce332e6b0be6b

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1;
   3: 1,    4;
   4: 1,   16,     18;
   5: 1,   44,    174,      96;
   6: 1,  104,    984,    1776,      600;
   7: 1,  228,   4356,   17600,    18540,     4320;
   8: 1,  480,  16830,  128480,   290700,   203904,    35280;
   9: 1,  988,  59850,  783200,  3220100,  4711824,  2387952,   322560;
  10: 1, 2008, 201780, 4250000, 28827500, 74640384, 77184072, 29871360, 3265920;

%F Formula

The row polynomials satisfy P(n,x) = -(4*n^2-18*n+18)*x^2*P(n-2,x) - ((2*n-3)*x^2-(2*n-3)*x^3)*d/dx P(n-2,x) + (1+(4*n-8)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 18*n + 18)*x^2*P[n - 2, x] - ((2*n - 3)*x^2 - (2*n - 3)*x^3)*D[P[n - 2, x], x] + (1 + (4*n - 8)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 18*n + 18)*x^2*v[i - 2] - ((2*n - 3)*x^2 - (2*n - 3)*x^3)*deriv(v[i - 2], x) + (1 + (4*n - 8)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Related ?

Same fingerprint ?

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.